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21 views

Efficiently compute the bases functions of the Signal [on hold]

Let there be 4 1D signals such that \begin{cases} x(t)=4\sin(10\pi t) \\ y(t)=8\cos(20\pi t) \\ z(t)=16\sin(30\pi t) \\ m(t)=x(t) +y(t)-z(t) \end{cases} Is there a way to compute ...
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1answer
878 views

DSP, discrete time, how to calculate the frequency

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b. The question is: Determine the fundamental period of $x[n] = cos(0.6n\pi + 0.3\pi)$ Since x[n], is in square brackets, ...
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1answer
8k views

Prove of the Parseval's theorem for Discrete Fourier Transform (DFT)

If $x[k]$ and $X[r] $ are the pair of discrete time Fourier sequences, where $x[k]$ is the discrete time sequence and $X[r]$ is its corresponding DFT. Prove that the energy of the aperiodic sequence ...
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68 views

Puzzle to puzzle you:image [on hold]

Suppose 3 1D Signals x(t), y1(t) and y2(t) are given as x(t)=sin(40*pi*t); y1(t)=.5*sin(40*pi*t) and y2(t)=x(t)+y1(t). Left side =Right side Here,values of x(t)and y1(t)i.e.(Right side) are given ...
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0answers
10 views

Is this solution correct? [Discrete-time signals and systems]

Consider this question taken from Oppenheim - Discrete-time Signal and Systems: Now consider its solution (from the Solutions manual) My question is: is the solution for itens (a) and (b) valid? ...
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0answers
17 views

A 2D smoothing convolution filter

I'm trying to find the right form of a 2D filter that will do the following to a matrix after linear convolution: Let A = [ ? ? ?] [ ? ? ?] [ ? ? ?] and B = ...
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0answers
19 views

Filter transfer function to state space

I'm trying to change this filter transfer function to state space representation $ y_t=\frac{1+b_1 z^{-1}}{1+a_1 z^{-1} +a_2 z^{-2}}u_t $ I tried writing it as time series $ y_t+a_1 ...
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2answers
69 views

Fourier Decompositon

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
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1answer
35 views

Calculating Fourier Transform of $\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$

This question deals with finding the Nyquist Frequency of a given signal. Suppose you have the signal $x(t)=\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$ in the time domain where $T>0$ is some ...
2
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2answers
615 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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1answer
327 views

Plot recursive signal in Matlab

I need to create and plot this signal in matlab with 2000 points: x(n) = 0.6530 x(n-1) - 0.7001 x(n-2) + v(n) Where $x(-1)=x(-2)=0$ and $v(n) =$ white noise I ...
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1answer
323 views

What is a moving average system?

Can someone elaborate on what a moving average system is? I know that the system is defined as: $$y[n] = \frac{x[n] + x[n-1] + x[n-2]}{3}$$ How would we draw $y[n]$ given that we have a graph with ...
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5answers
330 views

Compressive sensing with non square matrices

I'm implementing the algorithm in the following paper: "Compressive sensing for wideband cognitive radios" http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04218361 However I've run into a ...
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1answer
15 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
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0answers
12 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
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0answers
38 views

Phase vocoder equation

I know I can adjust the frequency of a waveform using a modified version of the sine wave equation $$\mathrm{amplitude}\times\cos(2\pi\times\mathrm{frequency}\times\mathrm{time}+\mathrm{phase})$$ ...
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1answer
26 views

If $f(t)$ is periodic, is there any $t$ that would equal to DC components?

Suppose $f(t)$ is periodic with period $T$. Would there be $t$ that would necessarily equal to DC component (it can be scaled)? By DC component, I mean $F(0)$ where $F$ is fourier coefficient of $f$. ...
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1answer
32 views

Is it possible to reconstruct signal using phase only or magnitude only?

I am studying Fourier Transform and it's inverse. We get phase and magnitude from Fourier transform and reconstruct it back from both together My question is that is it possible to reconstruct given ...
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1answer
23 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
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1answer
306 views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
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2answers
39 views

Discrete Fourier Transform of generalised Hamming Window

The generalised Hamming Window is defined as: $$ w(n) = \begin{cases} \alpha - (1 - \alpha)\cos(2 \pi n /N), & \text{if $ 0 \leq n \leq N$} \\ 0, & \text{otherwise} \end{cases} $$ with $ 0 ...
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1answer
35 views

What is the connection between random variables and time series?

I always felt that there was a disconnect between random variable and time series. Clearly, random variable and time series can both be treated with statistical methods. First order, second order ...
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1answer
1k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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0answers
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Does exise a,b,c make the same input lead to same output?

The first function is y[n]=x[n]+0.5*x[n-1]+0.25*x[n-2] the other is y[n]=a*x[n+1]+b*x[n]+c*x[n-1] Does exist some ...
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0answers
28 views

Fourier transform of a 3sinc^2(100πt)

I'm currently studying for an exam, and I'm not sure the textbook's answer for the fourier transform of 3sinc^2(60πt) is correct. For this question, I incorporated the duality property. Below is my ...
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0answers
13 views

How to express a signal in terms of Riesz bases?

Fast discrete wavelet transform allows us to express any discrete signal in terms of wavelet bases by convolution with filter coefficients. How can one express a digital signal in terms of ...
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2answers
8k views

How do I exactly project a vector onto a subspace?

I am trying to understand how - exactly - I go about projecting a vector onto a subspace. Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not ...
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1answer
41 views

Proof of the discrete Fourier transform of a discrete convolution

Let the discrete Fourier transform be $$ \mathcal{F}_N\mathbf{a}=\hat{\mathbf{a}},\quad \hat{a}_m=\sum_{n=0}^{N-1}e^{-2\pi i m n/N}a_n $$ and let the discrete convolution be $$ ...
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1answer
85 views

How one can show $P(ax+n|x)=P(n)$? [closed]

Let $x$ be a signal and $n$ be an independent noise. How one can show $P(ax+n|x)=P(n)$? Thanks. Well, let $y=ax+n$, so we have $n=y-ax$. Now if the probability density function (PDF) of $n$ for ...
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0answers
9 views

Errors of approximating continuous Fourier transform by discrete Fourier transform

In http://planetmath.org/approximatingfourierintegralswithdiscretefouriertransforms some error analysis of using DFT to approximate continuous Fourier transform is indeed done, but there are things I ...
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1answer
27 views

Show if signal is time variant or not

I know that I have to show that \begin{align*} y[n-n_0] &= f \Big( \{x[n - n_0]\} \Big) \end{align*} in order to tell if a signal is time-varying of not. Having a signal $y[n] = ...
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0answers
548 views

proof that orthogonal signals not interrupting each other

I know that signals that are orthogonal do not disturb each other. What I am curious is what is the proof behind why orthogonal signals in a single signal (i.e. a single signal can be broken down ...
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0answers
29 views

Exact formula for alias of Discrete Fourier transform for periodic sigals

Suppose that $f(t): \mathbb{R} \to \mathbb{C}$ is a $T$-periodic signal, with highest frequency $f_h$. Now suppose that our sampling rate frequency is lower than $f_h$, and is not any multiples of ...
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1answer
601 views

Calculate phase and amplitude of a sampled sine wave

I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two ...
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0answers
39 views

Real and imaginary part of an Eigenvector.

Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) generated by a band limited signal on a set of sensors. Is there a relation ...
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1answer
51 views

Is there a relation between half space and Eigenvectors?

I request earnestly apology if the question is not well defined. I think I understand half space and Eigenvectors to an extent, but could not connect both of them under the same geometry or ...
2
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0answers
85 views

Properties of eigenvectors of a sample covariance matrix?

My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the ...
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0answers
38 views

Savitzky-Golay Coefficients for end points

I've been looking for solution to clean up SG Filter end points and I discovered a shifted set of coefficients in Numerical Recipes that might do the trick. Nr = 0; Nl = 4; 0.086, -0.143, -0.086, ...
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0answers
20 views

Spectral Analysis: How to interpret a periodogram.

I'm reading a paper that has to do with financial volatility. The author uses a periodogram to estimate the power spectrum density of the volatility time-series. Evidently, the plot (below) is ...
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0answers
23 views

Condition Butterworth polynomial

My course states that a polynomial is a Butterworth polynomial when it satisfies the following condition: $|B(j\Omega)|=\sqrt {1+{\Omega}^{2\,n}}=\sqrt {1+{(\omega/\omega_p)}^{2\,n}}$ I'm really ...
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1answer
17 views

how can I plot the infinite sum in matlab [closed]

I'm lookin for a way to plot $$\hat x= \sum_{n=-\infty}^\infty 0.5cos(1.3\pi n)sinc(t-n)$$ in matlab, and I can't find out how
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0answers
22 views

A “Fourier Phase” for (stationary) random processes?

Let $X_t$ be a real w.s.s. random process. Its spectrum is given by $S(f)=\mathcal{F}R_X(\tau)(f)$ where $R_X$ is the process autocorrelation. As $X_t$ is real, the spectrum will be real and ...
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0answers
23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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1answer
16 views

Frequency scaling property for Fourier series

For Fourier transform, there is an equation connecting time-scaling with frequency-scaling. (By scaling, I mean multiplying by constant for time or frequency) Is there such a relation for Fourier ...
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1answer
21 views

Convergence property of DTFT toward DFT when function is periodic

from Wikipedia: When the input data sequence $x[n]$ is $N$-periodic, DTFT can be computationally reduced to a discrete Fourier transform (DFT), because: $ X_{1/T}(f)$ converges to zero ...
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1answer
32 views

Help in understanding step function calculation

Dear community I would appreciate if you can help me understand these equations. I mean how did he jump from line 1 to line 2? How do u[n] get cancel? Then in the last line where did the "8" come ...
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0answers
16 views

How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
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1answer
35 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
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1answer
78 views

$E[x_i^2 x_j^2]$ for white Gaussian noise

If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so $$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$ and $$E[x_n]=0$$ Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$ ...
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0answers
20 views

Is there anything similar to DTFT for Fourier series?

So if sampling condition is met well, with aperiodic signals we have discrete-time Fourier transform (DTFT) that allows us to get frequency-domain data that resemble continuous-time fourier transform. ...